Do shoppers need an account to take part in my Find & Earn gift hunt?

No account is needed. When a shopper clicks a hidden gift, a small form asks for their name and email. After they submit it, they see a message on screen with the result, for example the coupon they won, and they get an email too.

Each person who opens a gift is listed on the event's Submissions tab with their Name, Email, Item won, Completion won and Created At. Open it from Plugins → Find & Earn → Events by clicking Manage on the event.